Wednesday, April 29, 2015

Kazunori showed me this beautiful torus knot (1,2) he made a few days ago. This structure can be classified as a torus knot, or more specifically a twisted torus without knot at all. The space curve that this tubular structure approximates can be described by the parametric equations for the torus knot (q=1, p=2). Therefore, it is reasonable to call it as the torus knot (1,2).
To me, this structure seems to be a perfect example to show the influence of the particular operation, Vertical Shift, described in the following papers:
1. Chuang, C.; Jin, B.-Y. Torus knots with polygonal faces, Proceedings of Bridges: Mathematical Connections in Art, Music, and Science 2014, 59-64. pdf
2. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. Comments on Structural Types of Toroidal Carbon Nanotubes, J. Chin. Chem. Soc. 2013, 60, 949-954.
3. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
Another related operation is the Horizontal shift, which is not used in this structure.
Applying these two operations carefully (usually nontrivial), one can mimic
the bending and twisting of many space curves in an approximate way.